A convex lens (of focal length $20\mathrm{ }\mathrm{c}\mathrm{m}$ ) and a concave mirror, having their principal axes along the same lines, are kept $80\mathrm{ }\mathrm{c}\mathrm{m}$ apart from each other. The concave mirror is to the right of the convex lens. When an object is kept at a distance of $30\mathrm{ }\mathrm{c}\mathrm{m}$ to the left of the convex lens, its image remains at the same position even if the concave mirror is removed. The maximum distance of the object for which this concave mirror, by itself would produce a virtual image would be:

• A

  $30\mathrm{ }\mathrm{c}\mathrm{m}$
• B

  $25\mathrm{ }\mathrm{c}\mathrm{m}$
• C

  $20\mathrm{ }\mathrm{c}\mathrm{m}$
• Correct answer
  D

  $10\mathrm{ }\mathrm{c}\mathrm{m}$

Solution

Image formed by lens
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
$\Rightarrow \frac{1}{v}+\frac{1}{30}=\frac{1}{20}$
$\Rightarrow \frac{1}{v}=\frac{1}{20}-\frac{1}{30}$
$\Rightarrow v=60\mathrm{c}\mathrm{m}$
In the image remains at the same position even if the concave mirror is removed indicates that the image formed by lens is formed at the center of curvature of spherical mirror.
$\therefore$ Radius of curvature of mirror $=80-60=20\mathrm{c}\mathrm{m}$
Focal length, $f=10\mathrm{c}\mathrm{m}$
For virtual image to be formed, object is to be kept between focus and pole. Hence, the maximum distance of object from spherical mirror for which virtual image is formed will be its focal length 10cm

Hence, the correct answer is option: 'D'.